All of the 3rd grade teachers and students from Covington went on a field trip to an art museum. Tickets were $$5.50$ each for teachers and $$4.00$ each for students, and the group paid $$58.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$22.00$ each for teachers and $$11.50$ each for students, and the group paid $$191.50$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+4y = 58}$ ${22x+11.5y = 191.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-22x-16y = -232}$ ${22x+11.5y = 191.5}$ Add the top and bottom equations together. $ -4.5y = -40.5 $ $ y = \dfrac{-40.5}{-4.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {5.5x+4y = 58}$ to find $x$ ${5.5x + 4}{(9)}{= 58}$ $5.5x+36 = 58$ $5.5x = 22$ $x = \dfrac{22}{5.5}$ ${x = 4}$ You can also plug ${y = 9}$ into $ {22x+11.5y = 191.5}$ and get the same answer for $x$ ${22x + 11.5}{(9)}{= 191.5}$ ${x = 4}$ There were $4$ teachers and $9$ students on the field trips.